What is the 'meaning' of nowhere dense set?

Question

In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres).

So what is the 'meaning' (i.e motivation, intuitive/geometric meaning etc.) of nowhere dense set? Thank you.

Answer 1

A set $A \subseteq X$ is dense in $X$ if every element of $X$ is either in $A$ or a limit point of $A$. Hence the meaning of $A$ being nowhere dense is for any $x\in X$, there exists an open set $V$ containing $x$ such that $(V-\{x\})\cap A=\emptyset$. In a metric space, roughly speaking, this means at everywhere it is possible to choose a sufficiently small region so that it contains at most one point of $A$.

Answer 2

You can think of the word "nowhere" as meaning "in no open set". So a subset $A \subset X$ is nowhere dense if it is not dense in any open set, or more precisely: for each open subset $U \subset X$ the set $A \cap U$ is not a dense subset of $U$ (with respect to the subspace topology on $U$).

Answer 3

A set $A$ is nowhere dense if every nonempty open set contains a nonempty open set which is disjoint from $A.$